Quantum quenches in the non-integrable Ising modelbodri.elte.hu/seminar/kormos_20170215.pdf · with...

Quantum quenches in thenon-integrable Ising model

Márton Kormos“Momentum” Statistical Field Theory Group,

Hungarian Academy of SciencesBudapest University of Technology and Economics

in collaboration withTibor Rakovszky, Márton Mestyán, Gábor Takács, Mario Collura, Pasquale Calabrese

ELTE Particle Physics Seminar15/02/2017

Relaxation & thermalisation


• Do isolated quantum systems relax? Do they thermalise?


• Do isolated quantum systems relax? Do they thermalise?• What does this even mean?

we can’t distinguish the global state of our system from thermal equilibrium by performing local measurements.(The reduced density matrices of subsystems are mixed due to entanglement, the system acts as its own bath.)




• Integrable systems can’t thermalise: conserved quantities!




• Integrable systems can’t thermalise: conserved quantities!• Generalised Gibbs Ensemble:

GGE =e

Pm mQm

Tr eP

m mQm M. Rigol et al., PRL 98, 050405 (2007)





• What if integrability is broken? Is there a quantum KAM theorem?“Prethermalisation”

GGE =e

Pm mQm

Tr eP


G. Brandino, J.-S. Caux, R. Konik, PRX 5, 041043 (2015)






• Are there universal features of the time evolution?

GGE =e

Pm mQm

Tr eP








• Are there universal features of the time evolution? • Cold atom experiments!

A. Kaufman et al., Science 353, 794 (2016)

© 2006 Nature Publishing Group

A quantum Newton’s cradleToshiya Kinoshita1, Trevor Wenger1 & David S. Weiss1

It is a fundamental assumption of statistical mechanics that aclosed system with many degrees of freedom ergodically samplesall equal energy points in phase space. To understand the limits ofthis assumption, it is important to find and study systems that arenot ergodic, and thus do not reach thermal equilibrium. A fewcomplex systems have been proposed that are expected not tothermalize because their dynamics are integrable1,2. Some nearlyintegrable systems of many particles have been studied numeri-cally, and shown not to ergodically sample phase space3. However,there has been no experimental demonstration of such a systemwith many degrees of freedom that does not approach thermalequilibrium. Here we report the preparation of out-of-equili-brium arrays of trapped one-dimensional (1D) Bose gases, eachcontaining from 40 to 250 87Rb atoms, which do not noticeablyequilibrate even after thousands of collisions. Our results areprobably explainable by the well-known fact that a homogeneous1D Bose gas with point-like collisional interactions is integrable.Until now, however, the time evolution of out-of-equilibrium 1DBose gases has been a theoretically unsettled issue4–6, as practicalfactors such as harmonic trapping and imperfectly point-likeinteractions may compromise integrability. The absence of damp-ing in 1D Bose gases may lead to potential applications in forcesensing and atom interferometry.To see qualitatively why 1D gases might not thermalize, consider

the elastic collision of two isolated, identical mass classical particles inone dimension. Energy and momentum are conserved only if theysimply exchange momenta. Clearly, the momentum distribution of a1D ensemble of particles will not be altered by such pairwisecollisions. The well-known behaviour of Newton’s cradle (seeFig. 1a) is most easily understood in this way. Even when severalballs are simultaneously in contact, particles in an idealized Newton’scradle just exchange specific momentum values, though the expla-nation is more subtle7. Generalization of the Newton’s cradle toquantum mechanical particles lends it a ghostly air. Rather than justreflecting off each other, colliding particles can also transmit througheach other. When the particles are identical, the final states aftertransmission and reflection are indistinguishable.In general, correlations and overlap among 1D Bose gas wavefunc-

tions complicate the picture of independent particles colliding as in aNewton’s cradle. In fact, there are circumstances in which 1Dmomentum distributions are known to change in time. For example,when weakly coupled bosons are released from a trap, the conversionof mean field energy to kinetic energy changes the momentumdistribution. In the Tonks–Girardeau limit of infinite strengthinteractions8, although the 1D bosons interact locally like non-interacting fermions, their momentum distribution is not fermio-nic9,10. When a Tonks–Girardeau gas is released from a trap andexpands in one dimension, its momentum distribution evolves intothat of a trapped Fermi gas11–13. The quantum Newton’s cradle viewof particles colliding with each other and either reflecting ortransmitting can only be applied when the kinetic energy of thecollision greatly exceeds the energy per atom at zero temperature at

the prevailing density14. The collisions that we study satisfy thiscriterion well. Our observations extend from the Tonks–Girardeauregime, where only pairwise collisions can occur15, to the intermediatecoupling regime, where there can be three- (or more) body col-lisions15–17. In both regimes, atoms that are set oscillating and collidingin a trap do not appreciably thermalize during our experiment.We start our experiments with a Bose–Einstein condensate (BEC)

loaded into the combination of a blue-detuned two-dimensional(2D) optical lattice and a red-detuned crossed dipole trap (seeMethods). The combination of light trapsmakes a 2D array of distinct,parallel Bose gases, with the 2D lattice providing tight transverseconfinement and the crossed dipole trap providing weak axial trap-ping11. The dynamics within each tube of the 2D array are strictly 1Dbecause the lowest transverse excitation, "q r (where q r/2p ¼ 67 kHzis the transverse oscillation frequency), far exceeds all other energies in

LETTERS

Figure 1 |Classical and quantumNewton’s cradles. a, Diagram of a classicalNewton’s cradle. b, Sketches at various times of two out of equilibriumclouds of atoms in a 1D anharmonic trap,U(z). At time t ¼ 0, the atoms areput into a momentum superposition with 2"k to the right and 2"k to theleft. The two parts of the wavefunction oscillate out of phase with each otherwith a period t. Each atom collides with the opposite momentum grouptwice every full cycle, for instance, at t ¼ 0 and t/2. Anharmonicity causeseach group to gradually expand, until ultimately the atoms have fullydephased. Even after dephasing, each atom still collides with half the otheratoms twice each cycle.

1Department of Physics, The Pennsylvania State University, 104 Davey Laboratory, University Park, Pennsylvania 16802, USA.

Vol 440|13 April 2006|doi:10.1038/nature04693

900

T. Kinoshita et al., Nature 440, 900 (2006)

GGE =e

Pm mQm

Tr eP



What can we do?

• focus on 1D systems

What can we do?

• focus on 1D systems• free systems (bosons, fermions, anyons)

What can we do?

• focus on 1D systems• free systems (bosons, fermions, anyons)• systems that can be mapped to free models

(Ising or XY spin chain, hard core bosons, bosonisation) P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012)

What can we do?



• integrable systems: in some models the asymptotic state is understood via GGE or the Quench Action method(CFT, XXZ, Lieb-Liniger).

What can we do?




What can we do?

Time evolution is hard.




• lattice systems: efficient numerical methods based on MPS representations (tDMRG, iTEBD)

What can we do?





• lattice systems: efficient numerical methods based on MPS representations (tDMRG, iTEBD)

• continuum systems, field theory?universality out of equilibrium?

What can we do?


Outline of the talk

Outline of the talk

Part I: Truncated Hilbert space approach to non-equilibrium dynamics

with Tibor Rakovszky, Márton Mestyán, Mario Collura, Gábor Takács,Nucl. Phys. B 911, 805 (2016)

Outline of the talk

Part I: Truncated Hilbert space approach to non-equilibrium dynamics

Part II: Space-time structure of out of equilibrium evolution, dynamical confinement

with Tibor Rakovszky, Márton Mestyán, Mario Collura, Gábor Takács,Nucl. Phys. B 911, 805 (2016)

with Mario Collura, Pasquale Calabrese, Gábor Takács,Nature Physics, Advanced Online Publication

Part ITruncated Hilbert space method

Nucl. Phys. B 911, 805 (2016)

Truncated Hilbert Space ApproachV.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990)

H = H0 +Hpert


H = H0 +Hpert

• is solvable (CFT, free field theory) in a finite volume • the matrix elements of can be calculated in

the basis of truncated at some energy• the matrix of can be computed exactly

H0

Hpert

H

H0


H = H0 +Hpert



H0

Hpert

H

H0

Truncated Hilbert Space Approach

• diagonalization yields the spectrum and eigenstates; form factors, expectation values can be calculated

V.P. Yurov, Al. B. Zamolodchikov, Int. J. Mod. Phys. A 5, 3221 (1990)

H = H0 +Hpert



H0

Hpert

H

H0



• finite size and cutoff effects are well understoodRG for TCSA P. Giokas, G. Watts, arXiv 1106.2448, G. Feverati et al., J. Stat. Mech. P03011 (2008); R.M. Konik, Y. Adamov, Phys. Rev. Lett. 98, 147205 (2007); M. Lencsés, G. Takács, JHEP 09, 052 (2014)


H = H0 +Hpert



H0

Hpert

H

H0




• recent revival of the method: S. Rychkov, L. G. Vitale, Phys. Rev. D 91, 085011 (2015), Phys. Rev. D 93, 065014 (2016); Z. Bajnok, M. Lájer, JHEP 10, 050 (2016)


H = H0 +Hpert



H0

Hpert

H

H0


Potential for studying real time dynamics





H = H0 +Hpert



H0

Hpert

H

H0


Potential for studying real time dynamics• non-perturbative





H = H0 +Hpert



H0

Hpert

H

H0


Potential for studying real time dynamics• non-perturbative• does not rely on integrability: effects of integrability breaking!





H = H0 +Hpert



H0

Hpert

H

H0


Potential for studying real time dynamics• non-perturbative• does not rely on integrability: effects of integrability breaking!• access to microscopic data (spectrum, overlaps)





Quantum Ising chain and Ising field theory

HQIM = J

NX

i=1

x

i

x

i+1 + hz

NX

i=1

z

i

+ hx

NX

i=1

x

i

!


HQIM = J

NX

i=1

x

i

x

i+1 + hz

NX

i=1

z

i

+ hx

NX

i=1

x

i

!

For it can be mapped to free spinless fermions (Jordan-Wigner trf.)hx

= 0


Quantum critical point at separating the ferromagnetic and paramagnetic phases

hz

= 1, hx

= 0 hz < 1hz > 1

HQIM = J

NX

i=1

x

i

x

i+1 + hz

NX

i=1

z

i

+ hx

NX

i=1

x

i

!


= 0


J ! 1, hz

! 1, M = 2J |1 hz

|, h / J15/8hx

fixedScaling limit:


hz

= 1, hx

= 0 hz < 1hz > 1

HQIM = J

NX

i=1

x

i

x

i+1 + hz

NX

i=1

z

i

+ hx

NX

i=1

x

i

!


= 0


J ! 1, hz

! 1, M = 2J |1 hz

|, h / J15/8hx

fixedScaling limit:


hz

= 1, hx

= 0 hz < 1hz > 1

HIFT = Hc=1/2 +

Zdx "(x) + h

Zdx(x)

HIFT =

Zdx

1

2

i

2

(x)@

x

(x) (x)@x

(x) iM (x) (x)

+ h(x)

HQIM = J

NX

i=1

x

i

x

i+1 + hz

NX

i=1

z

i

+ hx

NX

i=1

x

i

!


= 0


J ! 1, hz

! 1, M = 2J |1 hz

|, h / J15/8hx

fixedScaling limit:


hz

= 1, hx

= 0 hz < 1hz > 1

HIFT = Hc=1/2 +

Zdx "(x) + h

Zdx(x)

HIFT =

Zdx

1

2

i

2

(x)@

x

(x) (x)@x

(x) iM (x) (x)

+ h(x)

HQIM = J

NX

i=1

x

i

x

i+1 + hz

NX

i=1

z

i

+ hx

NX

i=1

x

i

!


= 0

h = hM15/8

measure everything in M

` = ML

(x, t) =

r

L

X

n

e

n/2

pcosh

n

!a(

n

)e

ipnxiEnt+ !a

†(

n

)e

ipnx+iEnt,

¯

(x, t) = r

L

X

n

e

n/2

pcosh

n

!a(

n

)e

ipnxiEnt+ !a

†(

n

)e

ipnx+iEnt

a(n), a

†(n0) = n,n0

pn = M sinh n =

2n

L,

En = M cosh n

HFF = MX

n

cosh(n)a†(n)a(n)

|1, 2, ..., ni = a†(1)a†(2)...a

†(n)|0i

Truncated fermionic space

The mode expansion

diagonalises the Hamiltonian

acting on the Fock space

P. Fonseca, A. Zamolodchikov, J. Stat. Phys. 110, 527 (2003)

(x, t) =

r

L

X

n

e

n/2

pcosh

n

!a(

n

)e

ipnxiEnt+ !a

†(

n

)e

ipnx+iEnt,

¯

(x, t) = r

L

X

n

e

n/2

pcosh

n

!a(

n

)e

ipnxiEnt+ !a

†(

n

)e

ipnx+iEnt

a(n), a

†(n0) = n,n0

pn = M sinh n =

2n

L,

En = M cosh n

HFF = MX

n

cosh(n)a†(n)a(n)

|1, 2, ..., ni = a†(1)a†(2)...a

†(n)|0i

Truncated fermionic space

The mode expansion

diagonalises the Hamiltonian

acting on the Fock space

Truncation of the Hilbert space

MX

n

cosh(n)

P. Fonseca, A. Zamolodchikov, J. Stat. Phys. 110, 527 (2003)

Quantum quenches in the IFT


H(M0, 0) ! H(M, 0)Integrable quenches:



H(M0, 0) ! H(M,h 6= 0)Non-integrable quenches:


H(M, 0)Use the truncated basis of !







eiHt| 0i = J0(t)| 0i+ 21X

n=1

(i)nJn(t)Tn(H)| 0iTime evolution:

Tn+1(x) = 2xTn(x) Tn1(x) , T0(x) = 1 , T1(x) = xChebyshev polynomials:





eiHt| 0i = J0(t)| 0i+ 21X

n=1

(i)nJn(t)Tn(H)| 0iTime evolution:

Tn+1(x) = 2xTn(x) Tn1(x) , T0(x) = 1 , T1(x) = xChebyshev polynomials:

Maximal time: tmax

L/2

Benchmarking the method:Integrable quenches in the

ferromagnetic phase

H(M0, 0) ! H(M, 0)

P (W ) =X

↵

[W (E↵ E0)] |h 0|↵i|2

Statistics of work

ML = 40, = 14M

P (W ) =X

↵

[W (E↵ E0)] |h 0|↵i|2

Statistics of work

| 0i = N exp

(iX

n

K (n,M,M0) a†(n) a

†(n) |0i

)

K (,M,M0) = tan

1

2arctan (sinh ) 1

2arctan

M

M0sinh

N =

Y

n

1 + |K(n,M,M0)|2

1/2

ML = 40, = 14M

L (t) = |h 0|eiHt| 0i|2 =

Z

dWP (W )eiWt

2

Loschmidt echo

L (t) = |h 0|eiHt| 0i|2 =

Z

dWP (W )eiWt

2

Loschmidt echo

L(t) =

exp X

n

log

1 + |K(n,M,M0)|2e2iEnt

1 + |K(n,M,M0)|2

!

2

exact result:

h(t)i = et/for large t ,

=

2M

Z 1

0d|K()|2 sinh

1

+O(K6)

Decay of the magnetisation

analytic result for large time:

D. Schuricht, F. Essler, J. Stat. Mech. P04017 (2012)(P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012))

h(t)i = et/for large t ,

=

2M

Z 1

0d|K()|2 sinh

1

+O(K6)

Decay of the magnetisation

analytic result for large time:

D. Schuricht, F. Essler, J. Stat. Mech. P04017 (2012)(P. Calabrese, F. Essler, M. Fagotti, J. Stat. Mech. P07016, P07022 (2012))

1 1TFSA ATFSA

M0 = 0.5M 0.0336M 0.0334M 1.36 1.37M0 = 1.5M 0.0222M 0.0214M 1.36 1.37

Non-integrable quenches in the ferromagnetic phase

H(M0, 0) ! H(M,h 6= 0)

Spectrum at finite magnetic field

h = 0.1, = 8M


h = 0.1, = 8M

McCoy-Wu scenario of weak confinement

Free domain wall Bound state = meson

B. McCoy, T. Wu, PRD 18, 1259 (1978)


h = 0.1, = 8M

V (d) = h · d



B. McCoy, T. Wu, PRD 18, 1259 (1978)


h = 0.1, = 8M

V (d) = h · d

sinh (2#n) 2#n

= 2 (n 1/4) ,

mWKBn = 2M cosh (#n)

Ai(zn) = 0

mAin = M(2 + 2/3zn)

approximations for the meson spectrum



B. McCoy, T. Wu, PRD 18, 1259 (1978)

Time evolution of the magnetisation

M0 = 1.5M


M0 = 1.5M


M0 = 1.5M

hOiDE =X

↵

|C↵|2 h↵|O|↵iDiagonal ensemble


M0 = 1.5M

hOiDE =X

↵



M0 = 1.5M

hOiDE =X

↵


Oscillations around infinite time average with frequencies set by the meson masses.


M0 = 1.5M

hOiDE =X

↵


Oscillations around infinite time average with frequencies set by the meson masses.see also M.K., M. Collura, P. Calabrese, G. Takács, Nat. Phys. '16

Statistics of work and Loschmidt echo

M0 = 1.5M, ML = 40, h = 0.1, = 8M

Non-integrable quenches in the paramagnetic phase

H(M0, 0) ! H(M,h 6= 0)


h = 0.1, = 8M


h = 0.1, = 8M

The spectrum changes perturbatively, there is a single massive particle.



First order FF perturbation theory for pure quench:h

G. Delfino, J. Phys. A 47, 402001 (2014)J. Viti, G. Delfino, arXiv:1608.07612

h(t)i h2

M20

F 1,0(0|)

2[cos(Mt) + 1]


h(t)i = Aet/cos(ft) + C

For the combined quench it is damped! Let’s fit it with



h(t)i h2

M20

F 1,0(0|)

2[cos(Mt) + 1]






h(t)i h2

M20

F 1,0(0|)

2[cos(Mt) + 1]






h(t)i h2

M20

F 1,0(0|)

2[cos(Mt) + 1]

f = M(h)






h(t)i h2

M20

F 1,0(0|)

2[cos(Mt) + 1]

f = M(h)




2M

Z 1

0d|K()|2 sinh

1



h(t)i h2

M20

F 1,0(0|)

2[cos(Mt) + 1]

f = M(h)

Statistics of work and Loschmidt echo

M0 = 1.5M, ML = 40, h = 0.1, = 8M

Final check: comparison with iTEBD simulation on the lattice

HQIM = J

NX

i=1

x

i

x

i+1 + hz

NX

i=1

z

i

+ hx

NX

i=1

x

i

!

Conclusions (Part I)

Nucl. Phys. B 911, 805 (2016)


• demonstrated that Hamiltonian truncation methods can be used effectively to study non-equilibrium dynamics of field theories

Nucl. Phys. B 911, 805 (2016)



• non-integrable quench in the ferromagnetic phase: the low-lying mesons play an important role

Nucl. Phys. B 911, 805 (2016)




• non-integrable quench in the paramagnetic phase: oscillations, analytic formula

Nucl. Phys. B 911, 805 (2016)





• no prethermalisation

Nucl. Phys. B 911, 805 (2016)





• no prethermalisation• “rare states”? G. Biroli, C. Kollath, A. Läuchli, Phys. Rev. Lett. 105, 250401 (2010)

Nucl. Phys. B 911, 805 (2016)





• no prethermalisation• “rare states”? G. Biroli, C. Kollath, A. Läuchli, Phys. Rev. Lett. 105, 250401 (2010)

Outlook: quenches in the sine-Gordon model

Nucl. Phys. B 911, 805 (2016)

Part IIReal time confinement after a quench

in the quantum Ising chain

Nature Physics, Advanced online publicationarXiv:1604.03571

Part IIReal time confinement after a quench

in the quantum Ising chain

Small perturbations can have dramatic effects on the dynamics.

Nature Physics, Advanced online publicationarXiv:1604.03571

3

1

2

3

4

5

time

t

1

2

3

4

5

time

t

0 10 200

1

2

3

4

|i−j|

time

t

0 10 200

1

2

3

4

|i−j|

time

t

(d) V = 20(b) V = 2

0 0 (c) V = 5(a) V = 0

25−5

0

5x 10−30.005

0

−0.005

FIG. 1: (color online) Time evolution of the equal-time density correlation function Ci,j(t) of spinless fermions after a quenchfrom the CDW ground state of H(V0) with V0 = 10, evolved by the Hamiltonian H(V ), with (a) V = 0, (b) V = 2, (c) V = 5,(d) and V = 20.

to u(V = 0) = 2vF = 4th, as expected, where vF denotesthe Fermi velocity for V = 0. In addition to the lightcone, additional propagation fronts at later times can beidentified in Fig. 1(a), which, however, possess a lower ve-locity. This signals that slower quasiparticles stemmingfrom regions without linear dispersion also participatein spreading information. Figure 1(c) shows the evolu-tion of the correlation function for a quench within theCDW phase, i.e., a case which should not be describableby conformal field theory. Interestingly, we neverthelessfind a pronounced light-cone behavior in the correlationfunction. Although the conformal field theory underlyingthe treatment of Calabrese and Cardy is not valid in thisregion, the physical picture that ballistically propagatingquasiparticles are generated by the quench seems to hold.However, in contrast to the case of the quench to the LLdisplayed in Figs. 1(a) and (b), we see that a strong alter-nating pattern forms in the density correlation functionand remains present and qualitatively unchanged afterthe onset of the light cone.

A more detailed view of the temporal evolution of thecorrelation functions is shown in Fig. 2, in which we plotthe values of Ci,j(t) as a function of time for increasingdistance | i−j | for V = 0 and V = 2, the two extremes ofthe Luttinger-liquid phase. After the arrival of the firstsignal, oscillatory behavior as a function of time can beobserved at each distance. However, as V is increased,the observed oscillations both decrease in magnitude andare damped out more rapidly. Comparing the results forthe free case to the ones obtained for V = 2 in Fig. 2, itcan be seen that the incoming front travels with a highervelocity when V is larger, as can also be seen in Fig. 1.

In contrast to the oscillatory behavior in the Luttinger-liquid phase, a steady increase of the correlations is ob-served when the quench occurs within the CDW phase, ascan be seen in Fig. 3. The alternating pattern imprintedat the onset of the light cone is preserved. Presumably,the correlation functions saturate at some time that issignificantly longer than the maximum time reached here.While results for both V < V0 and V > V0 show the same

Manmana et al ’08: interacting fermionsCorrelations and entanglement after a quench in the Bose-Hubbard model 8

0.001

0.01

0.1

1

<b0b r>(

t)

0.01

0.1

<b0b r>(

t)lo

w p

ass r=2

r=3r=4r=5r=6

0 0.2 0.4 0.6 0.8 1 1.2 1.4time t [J-1]

-1

0

<n0n r>(

t)re

scal

ed

+

+

Figure 3. Time-evolution of correlation functions after a quench from Ui = 2J toUf = 40J . The upper panel shows the single particle correlation functions

b†0br

for dierent distances r. The correlations show partial revivals up to a time tr whenthey start to reach a quasi-steady state. This time tr grows approximately linearlywith the distance r as marked by the vertical lines. The central panel shows thesame correlations functions after filtering out the high frequencies, see text for details.The lowest panel shows the density density correlations function

n0nr

after shifting

and rescaling their amplitude for better visibility. The common vertical dashed linesdenote the arrival of the minima as determined from the density-density correlations.The data shown is ED for a L = 14 and DMRG data for L = 32 and filling n = 1.

correlations b†jbj+r and the density-density correlations njnj+r at equal time. In

Fig. 3 we show the time-evolution of the dierent correlations after a quench from the

superfluid, Ui = 2, to the Mott-insulating, Uf = 40, parameter regime.

Single-particle correlations The upper panel shows the correlations b†0br for dierent

distances r‡. For short times the single particle correlations oscillate with the period

2/Uf . The origin of these oscillations lies in the integer spectrum of the operator

nj(nj 1)/2. Consider the limit of very strong interactions, where the time-evolution

is totally dominated by the interactions. The time evolution of the single particle

correlations is given by

b†ibj(t) =

m,m0

mi,m0i+1 mj ,m0

j1 eiUf (m0jm0

i1)tcmcm0m|b†ibj|m.

Here we use the notation m for the Fock state with mi particles on site i. The time-

evolution of the correlation function is determined by the non-vanishing cross terms

‡ To extract these correlations from the DMRG data with open boundary conditions the average overcentral sites is taken. Note that for periodic boundary conditions this quantitiy is real due to symmetry,whereas for open boundary conditions an imaginary part can develop. However for the shown functionsand times the imaginary part is negligible.

Kollath-Lauechli ’08: Bose-Hubbard

t2

bosons described by the Bose-Hubbard Hamiltonian

H (U) = ÂhR,R0i

b†

RbR0 +h.c.+

U2 Â

RnR(nR 1), (3)

where R denotes a lattice site, hR,R0i a pair of nearest-neighbor sites, b†

R (bR) the creation (annihilation) operator ofa boson on site R, nR = b†

RbR the boson density on site R,and U the two-body interaction strength. In the following, thelattice will be either a 1D chain or a 2D square lattice, withperiodic boundary conditions and average density hnRi = 1.The system is first prepared in the ground state of H (Ui). Attime t = 0, it is then driven out of equilibrium upon realizing asudden quantum quench in the interaction strength, from Ui toUf. We study the dynamics of the density-density correlationfunction

N(R, t) = hnR(t)n0(t)ihnR(0)n0(0)i, (4)

where the average is over the ground state of H (Ui) and thedensity operators are evolved in time with H (Uf) i.e., Eq. (2)where both A and B are the density operators.

Our analysis makes use of the t-VMC approach [13] that webriefly outline here. The starting point is to define a class oftime-dependent variational many-body wave functions, whichwe take of the Jastrow type

Y(x, t) hx|Y(t)i= expÂr

ar(t)Or(x)

F0(x), (5)

where x spans a configuration basis, F0(x) is a bosonic time-independent state, and ar(t) are complex variational parame-ters coupled to a set of operators Or that are diagonal in the x-basis, i.e., hx|Or|x0i = dx,x0Or(x). The explicit form of theseoperators and their total number define the variational sub-space. Here we use the Fock basis, x = ni, and the completeset of density-density correlations, Or = ÂR nRnR+r, wherer spans all independent distances on the lattice. The initialstate is chosen to be the variational Jastrow ground state ofH (Ui) with |F0i the noninteracting-boson ground state ofH (0). This choice provides an excellent approximation ofthe exact ground state of H (Ui) [14, 15]. For instance, thesuperfluid-insulator transition is obtained for Uvar

c ' 5 andUvar

c ' 21 in 1D and 2D respectively, in fair agreement withexact results [16, 17].

The variational dynamics of the system is fully containedin the trajectories of the variational parameters ar(t). Thelatter are obtained by minimizing the Hilbert-space distancebetween the infinitesimal exact dynamics and the time deriva-tive of the variational state (5) at each time step. This processis equivalent to project the exact time-evolved wave functiononto the variational subspace. It yields a closed set of coupledequations of motion:

iÂr0

Sr,r0(t)...ar0(t) = hOrH it hOrithH it , (6)

where Sr,r0(t) = hOrOr0 it hOrithOr0 it and the quantum av-erages are taken over the time-dependent variational state (5).

t

R0

2

4

6

8

10

15 30 45 0

0.05

0.1

0.15

0.2

0.25

0.3(a)

N(R,t

)

t0 2 4 6 8 10

(b)R = 25

R = 20

R = 15

R = 10

R = 5

t

rhH2ithHi2thH2i0hHi20

Uf = 3Uf = 4Uf = 5

0.1 1 10 1000.8

1

1.2

1.4(c)

v ins

t

R

4.64.8

55.2

0 8 16 24

Figure 1: (Color on-line) Spreading of correlations in a 1D chain.(a) Density-density correlations N(R, t) versus separation and timefor a quench in the interaction strength from Ui = 2 to Uf = 4. Theinset shows the instantaneous velocity as obtained from t-VMC (redpoints) and exact diagonalization (for a 12-site lattice; blue point).(b) Time dependence of N(R, t) for various values of R. For clar-ity, the curves are vertically shifted by a value proportional to R, andthe linear light-cone wave-front clearly appears. (c) Relative energyfluctuations versus time for various values of Uf. The t-VMC calcu-lations are performed for 200 (a and b) or 500 (c) sites.

At each time, the quantum averages appearing in Eq. (6) arecomputed by variational Monte Carlo simulations and the lin-ear system of equations (6) is solved for

...ar(t). The trajec-

tories ar(t) are then found by time-integrating the functions...ar(t).

We emphasize that our variational scheme is symplectic andexactly conserves both the total energy and the square modu-lus of the wave function. In the numerical calculations, we usea sufficiently small time-step, d t = 0.01, and a fourth-orderRunge-Kutta integration scheme, which conserves the energywith a very small systematic error of the order of one part ina thousand, for times up to t = 100. The t-VMC is thereforeintrinsically stable, amenable to simulating time scales thatexceed by about two orders of magnitudes those achievableby t-DMRG in 1D, and applies as well in higher dimensions.

Results.— Let us first discuss our results for the 1D chain.Figure 1(a) shows the density-density correlation N(R, t) as afunction of separation and time for a quantum quench fromUi = 2 to Uf = 4. Figure 1(b) shows vertical cuts of the lat-ter, plotted with a vertical shift proportional to R for clarity.A light-cone effect is clearly visible: N(R, t) is unaffected atshort times, then develops a maximum at a finite time t?(R),and finally undergoes damped oscillations. Similar results are

Carleo et al., ’14: Bose-Hubbard 2

0 1 2 3 4 5 6 715

10

5

0

5

10

15

0 -0.015

FIG. 2. Space-time plot of the Sz correlation functions (3) for thequench from i = 4 to f = cos(/4). This particular value of thefinal interaction is chosen due to technical reason in the Bethe ansatzcalculations. The upper panel shows ground state data whereas thelower panel shows data from a thermal density matrix at T/J = 1.This illustrates that the light-cone effect in this observable persistsalso at finite temperatures.

product state (MPS) framework. We come back to the de-scription of the algorithm and a discussion of its performancetowards the end of this paper.

Results.— In the following we consider quenches to thespin-1/2 Heisenberg XXZ chain with anisotropy

H() = JL1X

i=1

Sx

i Sxi+1 + Sy

i Syi+1 + Sz

i Szi+1

. (1)

Initially, the system is prepared in a Gibbs state correspondingto an XXZ Hamiltonian with anisotropy i at a temperatureT , i.e.

(t = 0) = Z1 exp[H(i)] , =

1

kBT, (2)

where Z = Tr exp[H(i)] (we set kB = 1). Theanisotropy is then quenched at time t = 0+ from i to0 f 1, as depicted in Fig. 1(a), and the system sub-sequently evolves unitarily with Hamiltonian H(f ) [40]. Inorder to probe the spreading of correlations we consider thelongitudinal spin correlation functions

Sz(j; t) = hSzL/2(t)S

zj (t)i hSz

L/2(t)ihSzj (t)i (3)

centered around the middle of the chain. Results for Sz(j; t)are most easily visualized in space-time plots, and typical re-sults are shown in Fig. 2. The most striking feature observedin these plots is the light-cone effect: at a given separation jconnected correlations Sz(j; t) arise fairly suddenly at a timethat scales linearly with j.

These results demonstrate that the light-cone effect persistsfor mixed initial states, although the visibility of the signalis diminished with increasing temperature (until it vanishedcompletely at = 0 since the initial density matrix is trivial

0

0.5

1

0 2 40 5 10 15 200

2

4

6

8

10a) b)

FIG. 3. a) Extracted inflection points versus distance for differentinitial temperatures for the quench from = 4 to cos(/4). Thestraight lines correspond to the velocities extracted from the GGEwhere only the offset of the time axis has been fitted. The orangedashed line denotes the ground state Bethe ansatz velocity at f .b) Rescaled averaged spin correlation functions for the quench from = 4 to cos(/4) for T/J = 1 and the ground state (dashed line)and different distances j = 3, 5, 7 and 9. We omit the error bars forclarity of the figure. The time axis is relative to the first inflectionpoint of the correlation functions for j = 3. One can see that thesignal is delayed as the initial temperature is increased.

and stationary). Comparing the time evolution of the corre-lation functions for different initial temperatures, we see (cfFig. 2 and Fig. 3) that the signal front is delayed when thetemperature of the initial state is increased, signalling that thespreading slows down. We further observe that the spreadingvelocity is sensitive to the strength of the quench, i.e. the valueof the initial interaction. At this point we should note that thisfinding is unexpected. Based on our current understanding ofquenches to CFTs or of Lieb-Robinson bounds, there are nopredictions available which support spreading velocities de-pending on the initial state.

Having established the result that the spreading velocity de-pends both on the initial density matrices and the final Hamil-tonian, an obvious question is which properties of (t = 0)are relevant in this context. In order to quantify this aspect wedefine the precise location of the light-cone as the first inflec-tion point of the signal front observed in Sz (alike Ref. 29).This allows us to extract a spreading velocity vs by perform-ing a linear fit to the largest accessible time, where expectedfinite-distance effects [41] are small.

Our main result, shown in Fig. 4, is that the spreading ve-locity is mainly determined by the final energy density

ef =Tr[H(f )(t = 0)]

L. (4)

Plotting the measured velocities against ef leads to a remark-able data collapse for a variety of quenches from thermal aswell as pure initial states for various i. This holds in spiteof the fact that the system is integrable and thus its dynam-ics is constrained by an infinite set of conserved quantities.As we will show in the following, the observed velocities can

Bonnes, Essler, Lauchli ’14: XXZ spin chain

Light cone in interacting models

2

~~

position

time

b

d = v t

aquench

FIG. 1. Spreading of correlations in a quenched atomicMott insulator. a, A 1d ultracold gas of bosonic atoms(black balls) in an optical lattice is initially prepared deepin the Mott-insulating phase with unity filling. The latticedepth is then abruptly lowered, bringing the system out ofequilibrium. b, Following the quench, entangled quasiparticlepairs emerge at all sites. Each of these pairs consists of adoublon (red ball) and a holon (blue ball) on top of the unity-filling background, which propagate ballistically in oppositedirections. It follows that a correlation in the parity of thesite occupancy builds up at time t between any pair of sitesseparated by a distance d = vt, where v is the relative velocityof the doublons and holons.

mentum k, respectively, and k belongs to the first Bril-louin zone. Quasiparticles thus emerge at any site in theform of entangled pairs, consisting of a doublon and aholon with opposite momenta. Some of these pairs arebound on nearest-neighbour sites while the others formwave packets, due to their peaked momentum distribu-tion. The wave packets propagate in opposite directionswith a relative group velocity v determined by the dis-persion relation d(k) + h(k) of doublons and holons(Fig. 1b). The propagation of quasiparticle pairs is re-flected in the two-point parity correlation functions [21]:

Cd(t) = sj(t)sj+d(t) sj(t)sj+d(t) , (2)

where j labels the lattice sites. The operator sj(t) =ei[nj(t)n] measures the parity of the occupation numbernj(t). It yields +1 in the absence of quasiparticles (oddoccupancy) and -1 if a quasiparticle is present (even occu-pancy). Because the initial state is close to a Fock statewith one atom per lattice site, we expect Cd(t = 0) 0.After the quench, the propagation of quasiparticle pairswith the relative velocity v results in a positive correla-tion between any pair of sites separated by a distanced = vt.

The experimental sequence started with the prepara-tion of a two-dimensional (2d) degenerate gas of 87Rbconfined in a single antinode of a vertical optical lattice[17, 21] (z-axis, alat = 532nm). The system was thendivided into about 10 decoupled 1d chains by adding asecond optical lattice along the y-axis and by setting both

lattice depths to 20.0(5)Er, where Er = (2~)2/(8ma2lat)is the recoil energy of the lattice and m the atomic mass of87Rb. The eective interaction strength along the chainswas tuned via a third optical lattice along the x-axis. Thenumber of atoms per chain ranged between 10 and 18, re-sulting in a lattice filling n = 1 in the Mott-insulating do-main. The inital state was prepared by adiabatically in-creasing the x-lattice depth until the interaction strengthreached a value of (U/J)0 = 40(2). We then brought thesystem out of equilibrium by lowering the lattice depthtypically within 100 µs, which is fast compared to theinverse tunnel coupling ~/J , but still adiabatic with re-spect to transitions to higher Bloch bands. The finallattice depths were in the Mott-insulating regime, closeto the critical point. After a variable evolution time, we“froze” the density distribution of the many-body stateby rapidly raising the lattice depth in all directions to 80Er. Finally, the atoms were detected by fluorescenceimaging using a microscope objective with a resolutionon the order of the lattice spacing and a reconstructionalgorithm extracted the occupation number at each lat-tice site [17]. Because inelastic light-assisted collisionsduring the imaging lead to a rapid loss of atom pairs, wedirectly detected the parity of the occupation number.

Our experimental results for the time evolution of thetwo-point parity correlations after a quench to U/J =9.0(3) show a clear positive signal propagating with in-creasing time to larger distances d (Fig. 2). In addition,the propagation velocity of the correlation signal is con-stant over the range 2 d 6 (inset of Fig. 2). We foundsimilar dynamics also for quenches to U/J = 5.0(2) and7.0(3) (Fig. 4). We note that the observed signal can-not be attributed to a simple density wave because suchan excitation would result in sj sj+d = sjsj+d. Wecompared the experimental results to numerical simula-tions of an infinite, homogeneous system at T = 0 usingthe adaptive time-dependent density matrix renormal-ization group [22, 23] (t-DMRG). In the simulation, theinitial and final interaction strengths were fixed at the ex-perimentally determined values and the quench was con-sidered instantaneous, at t = 0. We found remarkableagreement between the experiment and theory over allexplored distances and times, despite the finite tempera-ture T 0.1U/kb (kb is the Boltzmann constant) and theharmonic confinement with frequency = 68(1)Hz thatcharacterise the experimental system. The observed dy-namics is also qualitatively reproduced by our analyticalmodel for U/J = 9.0. For lower values of U/J , however,the model breaks down due to the increasing number ofquasiparticles.

We extracted the propagation velocity v from the timeof the correlation peak as a function of the distanced (Fig. 3a). A linear fit restricted to 2 d 6yields v ~/(Jalat) = 5.0(2), 5.6(5) and 5.0(2) for U/J =5.0(2), 7.0(3) and 9.0(3), respectively. The points ford = 1 were excluded from the fit, as they result from the

3

FIG. 2. Time evolution of the two-point parity cor-relations. After the quench, a positive correlation signalpropagates with increasing time to larger distances. The ex-perimental values for a quench from U/J = 40 to U/J = 9.0(circles) are in good agreement with the corresponding numer-ical simulation for an infinite, homogeneous system at zerotemperature (continuous line). Our analytical model (dashedline) also qualitatively reproduces the observed dynamics. In-set: Experimental data displayed as a colormap, revealing thepropagation of the correlation signal with a well defined ve-locity. The experimental values result from the average overthe central N sites of more than 1000 chains, where N equals80% of the length of each chain. Error bars represent thestandard deviation.

interference between propagating and bound quasiparti-cle pairs (Eq. (1)). A comparison of the experimentalvelocities with the ones obtained from numerical simu-lations (Fig. 3b) shows agreement within the error bars.The measured velocities can also be compared with twolimiting cases: On the one hand, they are significantlylarger than the spreading velocity of non-interacting par-ticles, v = 4 Jalat/~, and twice the velocity of soundin the superfluid phase [24]; on the other hand, they re-main below the maximum velocity predicted by our eec-tive model, that can be interpreted as a Lieb–Robinson

FIG. 3. Propagation velocity. a, Determination of thepropagation velocity for the quenches to U/J = 5.0, 7.0 and9.0. The time of the maximum of the correlation signal isobtained from fits to the traces Cd(t) (circles). Error barsrepresent the 68% confidence interval of these fits. We thenextract the propagation velocities from weigthed linear fitsrestricted to 2 d 6 (lines). The data for U/J = 5.0 and7.0 have been oset horizontally for clarity. b, Comparisonof the experimental velocities (circles) to the ones obtainedfrom numerical simulations for an infinite, homogeneous sys-tem at zero temperature (shaded area). The shaded area andthe vertical error bars denote the 68 % confidence interval ofthe fit. The horizontal error bars represent the uncertaintydue to the calibration of the lattice depth. The black line cor-responds to the bound predicted by our eective model (theshading indicates the break down of this model). The arrowsmark the maximum velocity expected in the non-interactingcase (left) and the asymptotic value derived from our modelwhen U/J ! 1 (right).

bound (Fig. 3b). This bound equals 6 Jalat/~ in the limitU/J , corresponding to doublons and holons propa-gating with the respective group velocities 4 Jalat/~ and2 Jalat/~. The higher velocity of doublons simply reflectstheir Bose-enhanced tunnel coupling.

In conclusion, we have presented the first experimen-tal observation of an eective light cone for the spread-ing of correlations in an interacting quantum many-bodysystem. Although the observed dynamics can be under-stood within a fermionic quasiparticle picture valid deep

M. Cheneau et al., Nature 481, 484 (2012)

Light cone in experiments

P. Calabrese, J. Cardy 2005

Light cone spreading of entanglement entropy


• After a global quench, the initial state has an extensive excess of energy


| 0i

t

2t 2t

l

t

2t > l

2t < l

AB B

ABB

FIG. 7. Space-time picture illustrating how the entanglement between an interval A and therest of the system, due to oppositely moving coherent quasiparticles, grows linearly and thensaturates. The case where the particles move only along the light cones is shown here for clarity.

momentum p produced at x is therefore at x + v(p)t at time t, ignoring scattering effects.Now consider these quasiparticles as they reach either A or B at time t. The field at

some point x′ ∈ A will be entangled with that at a point x′′ ∈ B if a pair of entangledparticles emitted from a point x arrive simultaneously at x′ and x′′ (see Fig. 7).

The entanglement entropy between x′ and x′′ is proportional to the length of the intervalin x for which this can be satisfied. Thus the total entanglement entropy is

SA(t) ≈!

x′∈A

dx′

!

x′′∈B

dx′′

! ∞

−∞

dx

!

f(p′, p′′)dp′dp′′δ"

x′ − x − v(p′)t#

δ"

x′′ − x − v(p′′)t#

.

(4.1)Now specialize to the case where A is an interval of length ℓ. The total entanglement

is twice that between A and the real axis to the right of A, which corresponds to takingp′ < 0, p′′ > 0 in the above. The integrations over the coordinates then give max

"

(v(−p′) +v(p′′))t, ℓ

#

, so that

SA(t) ≈ 2t

! 0

−∞

dp′! ∞

0

dp′′f(p′, p′′)(v(−p′) + v(p′′)) H(ℓ − (v(−p′) + v(p′′))t) +

+ 2ℓ

! 0

−∞

dp′! ∞

0

dp′′f(p′, p′′) H((v(−p′) + v(p′′))t − ℓ) , (4.2)

15



• It acts as a source of quasi-particles at . A particle of momentum p has energy and velocity


| 0i

t = 0vp = dEp/dpEp

t

2t 2t

l

t

2t > l

2t < l

AB B

ABB





SA(t) ≈!

x′∈A

dx′

!

x′′∈B

dx′′

! ∞

−∞

dx

!


x′ − x − v(p′)t#

δ"

x′′ − x − v(p′′)t#

.



"

(v(−p′) +v(p′′))t, ℓ

#

, so that

SA(t) ≈ 2t

! 0

−∞

dp′! ∞

0


+ 2ℓ

! 0

−∞

dp′! ∞

0

dp′′f(p′, p′′) H((v(−p′) + v(p′′))t − ℓ) , (4.2)

15




• For the particles move semiclassically with velocity


| 0i

t = 0vp = dEp/dpEp

t > 0 vp

t

2t 2t

l

t

2t > l

2t < l

AB B

ABB





SA(t) ≈!

x′∈A

dx′

!

x′′∈B

dx′′

! ∞

−∞

dx

!


x′ − x − v(p′)t#

δ"

x′′ − x − v(p′′)t#

.



"

(v(−p′) +v(p′′))t, ℓ

#

, so that

SA(t) ≈ 2t

! 0

−∞

dp′! ∞

0


+ 2ℓ

! 0

−∞

dp′! ∞

0

dp′′f(p′, p′′) H((v(−p′) + v(p′′))t − ℓ) , (4.2)

15





• Particles emitted from regions of size of the initial correlation length are correlated and entangled, particles from points far away are incoherent


| 0i

t = 0vp = dEp/dpEp

t > 0 vp

t

2t 2t

l

t

2t > l

2t < l

AB B

ABB





SA(t) ≈!

x′∈A

dx′

!

x′′∈B

dx′′

! ∞

−∞

dx

!


x′ − x − v(p′)t#

δ"

x′′ − x − v(p′′)t#

.



"

(v(−p′) +v(p′′))t, ℓ

#

, so that

SA(t) ≈ 2t

! 0

−∞

dp′! ∞

0


+ 2ℓ

! 0

−∞

dp′! ∞

0

dp′′f(p′, p′′) H((v(−p′) + v(p′′))t − ℓ) , (4.2)

15





• Particles emitted from regions of size of the initial correlation length are correlated and entangled, particles from points far away are incoherent


| 0i

t = 0vp = dEp/dpEp

t > 0 vp

• When is bounded (e.g. Lieb-Robinson bounds) , the entanglement entropy grows linearly with time up to a value linear in

vp |vp| < vmax

`

Example: Transverse Field Ising chainP. Calabrese, J. Cardy 2005

Example: Transverse Field Ising chain

Evolution of entanglement entropy following a quantum quench:

Analytic results for the XY chain in a tranverse magnetic field

Maurizio Fagotti and Pasquale CalabreseDipartimento di Fisica dell’Universita di Pisa and INFN, Pisa, Italy

(Dated: November 28, 2010)

The non-equilibrium evolution of the block entanglement entropy is investigated in the XY chainin a transverse magnetic field after the Hamiltonian parameters are suddenly changed from and toarbitrary values. Using Toeplitz matrix representation and multidimensional phase methods, weprovide analytic results for large blocks and for all times, showing explicitly the linear growth intime followed by saturation. The consequences of these analytic results are discussed and the e↵ectsof a finite block length is taken into account numerically.

PACS numbers: 03.67.Mn, 02.30.Ik, 64.60.Ht

The non-equilibrium evolution of extended quantumsystems is one of the most challenging problems of con-temporary research in theoretical physics. The subjectis in a renaissance era after the experimental realization[1] of cold atomic systems that can evolve out of equilib-rium in the absence of any dissipation and with high de-gree of tunability of Hamiltonian parameters. A stronglylimiting factor for a better understanding of these phe-nomena is the absence of e↵ective numerical methods tosimulate the dynamics of quantum systems. For meth-ods like time dependent density matrix renormalizationgroup (tDMRG) [2] this has been traced back [3] to a toofast increasing of the entanglement entropy between partsof the whole system and the impossibility for a classicalcomputer to store and manipulate such large amount ofquantum information.

This observation partially moved the interest from thestudy of local observables to the understanding of theevolution of the entanglement entropy and in particularto its growth with time. Based on early results fromconformal field theory [5, 6] and on exact/numerical onesfor simple solvable model [5, 7] it is widely accepted [3]that the entanglement entropy grows linearly with timefor a so called global quench (i.e. when the initial statedi↵ers globally from the ground state and the excess ofenergy is extensive), while at most logarithmically for alocal one (i.e. when the the initial state has only a localdi↵erence with the ground state and so a little excess ofenergy). As a consequence a local quench is simulable bymeans of tDMRG, while a global one is not.

However, despite this fundamental interest and a largee↵ort of the community, still analytic results are lacking.In this letter we fill this gap providing the full analyticexpression for the entanglement entropy at any time inthe limit of large block for the XY chain in a transversemagnetic field. The model is described by the Hamilto-nian

H(h, ) = NX

j=1

1 +

4x

j

x

j+1 +1

4y

j

y

j+1 +h

2z

j

,

(1)

where ↵

j

are the Pauli matrices at the site j. Periodicboundary conditions are always imposed. Despite of itssimplicity, the model shows a rich phase diagram beingcritical for h = 1 and any and for = 0 and h 1, withthe two critical lines belonging to di↵erent universalityclasses. The block entanglement entropy is defined as theVon Neumann entropy S

`

= Tr`

log `

, where `

=Tr

n`

is the reduced density matrix of the block formedby ` contiguous spins. In the following we will considerthe quench with parameters suddenly changed at t = 0from h0, 0 to h, .

Our result is that, in the thermodynamic limit N !1and subsequently in the limit of a large block ` 1, thetime dependence of the entanglement entropy is

S(t) = t

Z

2|0|t<`

d'

22|0|H(cos

'

)+ `

Z

2|0|t>`

d'

2H(cos

'

) ,

(2)where 0 = d/d' is the derivative of the dispersion re-lation 2 = (h cos ')2 + 2 sin2 ' and represents themomentum dependent sound velocity (that because oflocality has a maximum we indicate as v

M

max'

|0|),cos

'

= (hh0 cos '(h + h0) + cos2 ' + 0 sin2 ')/0contains all the quench information [8] and H(x) =((1 + x)/2 log(1 + x)/2 + (1 x)/2 log(1 x)/2).

We first prove Eq. (2) and then discuss its interpreta-tion and physical consequences. The readers not inter-ested to the derivation can jump directly to latter part.

The method. Writing the entanglement entropy interms of a block Toeplitz matrix is rather standard[5, 9]. One first introduce Majorana operators a2l1 Q

m<l

z

m

x

l

and a2l

Q

m<l

z

m

y

l

and the corre-lation matrix A

`

through the relation ham

an

i = mn

+iA

`

mn

with 1 m, n `, that is a block Toeplitz matrix

`

=

2

66664

0 1 · · · `1

1 0

......

. . ....

1`

· · · · · · 0

3

77775,

l

=f

l

gl

gl

fl

.

Analytically for with constant

The determination of the time-dependent state | (t)i = eiHI(h)t| 0

i (and consequently

of the entanglement entropy) proceeds with the Jordan-Wigner transformation in terms of

Dirac or Majorana fermionic operators. All the details of these computations can be found

in the Appendix A.

The final result is that the time-dependent entanglement entropy for ` consecutive spins

in the chain can be obtained (analogously to the ground state case [2]) from the correlation

matrix of the Majorana operators

a2l1

Y

m<l

zm

!xl , a

2l Y

m<l

zm

!yl . (3.2)

We introduce the matrix A` through the relation hamani = mn + iA

` mn with 1 m,n `.

It has the form of a block Toeplitz matrix

A` =

2

6666664

0

1

· · · 1`

1

0

......

. . ....

`1

· · · · · · 0

3

7777775, l =

2

4 fl gl

gl fl

3

5 . (3.3)

with

gl =1

2

Z2

0

d'ei'lei'(cos' i sin' cos 2't) ,

fl =i

2

Z2

0

d'ei'l sin' sin 2't , (3.4)

where

' =q(h cos')2 + sin2 ' ,

0' =q(h

0

cos')2 + sin2 ' ,

ei' =cos' h i sin'

',

sin' =sin'(h

0

h)

'0',

cos' =1 cos'(h+ h

0

) + hh0

'0'. (3.5)

Calling the eigenvalues of A` as ±im, m = 1 . . . `, the entanglement entropy is S =

P`m=1

H(m) where H(x) is

H(x) = 1 + x

2log

1 + x

2 1 x

2log

1 x

2. (3.6)

8

The determination of the time-dependent state | (t)i = eiHI(h)t| 0

i (and consequently

of the entanglement entropy) proceeds with the Jordan-Wigner transformation in terms of

Dirac or Majorana fermionic operators. All the details of these computations can be found

in the Appendix A.

The final result is that the time-dependent entanglement entropy for ` consecutive spins

in the chain can be obtained (analogously to the ground state case [2]) from the correlation

matrix of the Majorana operators

a2l1

Y

m<l

zm

!xl , a

2l Y

m<l

zm

!yl . (3.2)

We introduce the matrix A` through the relation hamani = mn + iA

` mn with 1 m,n `.

It has the form of a block Toeplitz matrix

A` =

2

6666664

0

1

· · · 1`

1

0

......

. . ....

`1

· · · · · · 0

3

7777775, l =

2

4 fl gl

gl fl

3

5 . (3.3)

with

gl =1

2

Z2

0

d'ei'lei'(cos' i sin' cos 2't) ,

fl =i

2

Z2

0

d'ei'l sin' sin 2't , (3.4)

where

' =q(h cos')2 + sin2 ' ,

0' =q(h

0

cos')2 + sin2 ' ,

ei' =cos' h i sin'

',

sin' =sin'(h

0

h)

'0',

cos' =1 cos'(h+ h

0

) + hh0

'0'. (3.5)

Calling the eigenvalues of A` as ±im, m = 1 . . . `, the entanglement entropy is S =

P`m=1

H(m) where H(x) is

H(x) = 1 + x

2log

1 + x

2 1 x

2log

1 x

2. (3.6)

8

Evolution of entanglement entropy following a quantum quench:

Analytic results for the XY chain in a tranverse magnetic field

Maurizio Fagotti and Pasquale CalabreseDipartimento di Fisica dell’Universita di Pisa and INFN, Pisa, Italy

(Dated: November 28, 2010)

The non-equilibrium evolution of the block entanglement entropy is investigated in the XY chainin a transverse magnetic field after the Hamiltonian parameters are suddenly changed from and toarbitrary values. Using Toeplitz matrix representation and multidimensional phase methods, weprovide analytic results for large blocks and for all times, showing explicitly the linear growth intime followed by saturation. The consequences of these analytic results are discussed and the e↵ectsof a finite block length is taken into account numerically.

PACS numbers: 03.67.Mn, 02.30.Ik, 64.60.Ht

The non-equilibrium evolution of extended quantumsystems is one of the most challenging problems of con-temporary research in theoretical physics. The subjectis in a renaissance era after the experimental realization[1] of cold atomic systems that can evolve out of equilib-rium in the absence of any dissipation and with high de-gree of tunability of Hamiltonian parameters. A stronglylimiting factor for a better understanding of these phe-nomena is the absence of e↵ective numerical methods tosimulate the dynamics of quantum systems. For meth-ods like time dependent density matrix renormalizationgroup (tDMRG) [2] this has been traced back [3] to a toofast increasing of the entanglement entropy between partsof the whole system and the impossibility for a classicalcomputer to store and manipulate such large amount ofquantum information.

This observation partially moved the interest from thestudy of local observables to the understanding of theevolution of the entanglement entropy and in particularto its growth with time. Based on early results fromconformal field theory [5, 6] and on exact/numerical onesfor simple solvable model [5, 7] it is widely accepted [3]that the entanglement entropy grows linearly with timefor a so called global quench (i.e. when the initial statedi↵ers globally from the ground state and the excess ofenergy is extensive), while at most logarithmically for alocal one (i.e. when the the initial state has only a localdi↵erence with the ground state and so a little excess ofenergy). As a consequence a local quench is simulable bymeans of tDMRG, while a global one is not.

However, despite this fundamental interest and a largee↵ort of the community, still analytic results are lacking.In this letter we fill this gap providing the full analyticexpression for the entanglement entropy at any time inthe limit of large block for the XY chain in a transversemagnetic field. The model is described by the Hamilto-nian

H(h, ) = NX

j=1

1 +

4x

j

x

j+1 +1

4y

j

y

j+1 +h

2z

j

,

(1)

where ↵

j

are the Pauli matrices at the site j. Periodicboundary conditions are always imposed. Despite of itssimplicity, the model shows a rich phase diagram beingcritical for h = 1 and any and for = 0 and h 1, withthe two critical lines belonging to di↵erent universalityclasses. The block entanglement entropy is defined as theVon Neumann entropy S

`

= Tr`

log `

, where `

=Tr

n`

is the reduced density matrix of the block formedby ` contiguous spins. In the following we will considerthe quench with parameters suddenly changed at t = 0from h0, 0 to h, .

Our result is that, in the thermodynamic limit N !1and subsequently in the limit of a large block ` 1, thetime dependence of the entanglement entropy is

S(t) = t

Z

2|0|t<`

d'

22|0|H(cos

'

)+ `

Z

2|0|t>`

d'

2H(cos

'

) ,

(2)where 0 = d/d' is the derivative of the dispersion re-lation 2 = (h cos ')2 + 2 sin2 ' and represents themomentum dependent sound velocity (that because oflocality has a maximum we indicate as v

M

max'

|0|),cos

'

= (hh0 cos '(h + h0) + cos2 ' + 0 sin2 ')/0contains all the quench information [8] and H(x) =((1 + x)/2 log(1 + x)/2 + (1 x)/2 log(1 x)/2).

We first prove Eq. (2) and then discuss its interpreta-tion and physical consequences. The readers not inter-ested to the derivation can jump directly to latter part.

The method. Writing the entanglement entropy interms of a block Toeplitz matrix is rather standard[5, 9]. One first introduce Majorana operators a2l1 Q

m<l

z

m

x

l

and a2l

Q

m<l

z

m

y

l

and the corre-lation matrix A

`

through the relation ham

an

i = mn

+iA

`

mn

with 1 m, n `, that is a block Toeplitz matrix

`

=

2

66664

0 1 · · · `1

1 0

......

. . ....

1`

· · · · · · 0

3

77775,

l

=f

l

gl

gl

fl

.

t/lt, l 1M. Fagotti, P. Calabrese, 2008


The same scenario is valid for correlations:

Light cone spreading of correlations

• Horizon: points at separation become correlated when left- and right-moving particles originating from the same point first reach them



r


• If , connected correlations are then frozen for



|vp| < vmax

t < r/2vmax

r


• If , connected correlations are then frozen for


Quantum Quench in the Transverse Field Ising chain I 9

Figure 4. Numerical data for a quench within the ferromagnetic phase from h0

= 1/3 toh = 2/3. Left: The two-point function against the asymptotic prediction Eq. (19) for ` = 30 (upto a multiplicative factor) showing excellent agreement in the scaling regime. Inset: Ratio betweenthe numerical data and asymptotic prediction (69). The leading correction is time independent,but subleading contributions oscillate. Right: The connected correlation function for the sameparameters as on the left. For t < t

F

, xx

c

(`, t) vanishes identically in the scaling regime.

In the limit ` ! 1 (19) gives the square of the result (13) for the one-point function. Fortimes smaller than the Fermi time

tF =`

2vmax

, (21)

the first exponential factor in (19) equals 1. Thus, in the space-time scaling limit, connectedcorrelations vanish identically for times t < tF and begin to form only after the Fermi time. Thisis a general feature of quantum quenches [9, 22] and has been recently observed in experiments onone dimensional cold-atomic gases [4]. We stress that this by no means implies that the connectedcorrelations are exactly zero for t < tF : in any model, both on the lattice or in the continuumthere are exponentially suppressed terms (in `) which vanish in the scaling limit. The form factorapproach gives the following result for large t and ` (see Section 4.3)

xxFF (`, t) ' (1 h2)

1

4 exph

2`

Z

0

dk

K2(k)H

2"0h(k)t `

i

exph

4t

Z

0

dk

"0

h(k)K2(k)H

` 2"0h(k)t

i

. (22)

As expected, it gives the low density approximation to the full result (19).A comparison (for a typical quench from h

0

= 1/3 to h = 2/3) between the asymptotic results(19) (22) and numerical results for the correlation function at a finite but large distance (` = 30) isshown in Fig. 4. The numerical results are obtained by expressing the two-point correlator in thethermodynamic limit as the determinant of an ` ` matrix (see section 3) and then evaluating thedeterminant for di↵erent times. As we are concerned with equal time correlators only we do notneed to extract the two-point function from a cluster decomposition of the 4-point function [61].The agreement is clearly excellent. The ratio between the exact numerics and the analytic result(19) in the space-time scaling limit is shown in the inset of Fig. 4 for two values of = v

max

t/`.We see the ratio approaches a constant for large `. The corrections to this constant are seen to

Example: Ising model within ferromagnetic phaseP. Calabrese, F. Essler, M. Fagotti 2011/12


|vp| < vmax

t < r/2vmax

r

2

(a)

(b)

FIG. 1: Death of the light cone. Connected longitudinal spin-spin correlation function

hx

1

x

m+1

ic

after quenching to the ferromagnetic point hz

= 0 with a longitudinal magnetic field

hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j

are the Pauli matrices and we impose

periodic boundary conditions. The parameters

hz

and hx

are dimensionless, while J sets the en-

ergy scale. For hx

= 0 we recover the integrable

transverse field Ising model (TFIM) which can

be diagonalized by a Jordan-Wigner mapping to

free spinless Majorana fermions ak

[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)

with the dispersion relation "(k) =

2Jp1 2h

z

cos k + hz

2.

At hz

= 1 the system has a quantum criti-

cal point separating the paramagnetic and ferro-

magnetic phases. For hz

< 1 the system is in the

gapped ferromagnetic phase where the massive

fermions can be thought of as freely propagating

domain walls separating domains of magnetiza-

tion = (1 hz

2)1/8. Switching on a non-zero

field hx

induces a linear attractive potential be-

tween pairs domain walls which enclose a domain

of length d and of magnetization in the direction

Starting from the ferromagnetic state (all spins up) and evolving with

with hz = 0.25

Suppression of the light cone

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx





with hz = 0.25


2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx





with hz = 0.25


2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

h = 0h = 0.025h = 0.05h = 0.1h = 0.2h = 0.4

t

S

h = 0/0.5 → h = 0.25z

x

x

x

x

x

x

z

Entanglement entropy

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




McCoy & Wu ’78

Confinement in the Ising model

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




• For free fermions with dispersion

McCoy & Wu ’78


Dynamical confinement in the transverse field Ising chain

Marton Kormos

(Dated: December 5, 2015)

I. CONFINEMENT IN THE TRANSVERSE FIELD ISING CHAIN

The Hamiltonian that we intend to study is

H = JLX

j=1

x

j

x

j+1

+ hzz

j

+ hxx

j

, (1)

where ↵

j

are the Pauli matrices and we impose periodic boundary conditions. Note that hz and hx

are now dimensionless factors and J sets the energy scale. This Hamiltonian is sometimes referred

to as the “tilted field Ising chain”.

For hx = 0 we recover the integrable transverse field Ising chain which can be diagonalized by

mapping it to free spinless fermions:

HTI

=X

k

"(k)a†k

ak

+ const. , (2)

where the dispersion relation is given by

"(k) = 2Jp

1 2hz cos k + hz2 . (3)

Some care has to be taken with respect to the boundary conditions for the fermions and the

quantization of the momentum k, but we do not go into these details here.

At hz = 1 the system has a quantum critical point separating the paramagnetic and ferro-

magnetic phases. For hz < 1 the system is in the gapped ferromagnetic phase where the massive

fermions can be thought of as freely propagating domain walls separating domains of magnetization

= (1 hz2)1/8. This picture becomes more and more accurate as hz approaches zero.

A small (?) non-zero field hx induces a linear attractive potential between neighboring domain

walls which border a domain having magnetization in the direction opposite to hx. If d is the

distance between the domain walls, the potential is V (d) = · d with = 2Jhx. Clearly, domain

walls do not propagate freely anymore and they get confined into bound states (“mesons”).

hx

= 0

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx





• separates two massive phases

McCoy & Wu ’78



Marton Kormos




H = JLX

j=1

x

j

x

j+1

+ hzz

j

+ hxx

j

, (1)

where ↵

j






HTI

=X

k

"(k)a†k

ak

+ const. , (2)


"(k) = 2Jp

1 2hz cos k + hz2 . (3)











hx

= 0

hz = 1

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx






• For (ferro phase), the massive fermions can be seen as domain walls separating domains of magnetization

Free DW

McCoy & Wu ’78



Marton Kormos




H = JLX

j=1

x

j

x

j+1

+ hzz

j

+ hxx

j

, (1)

where ↵

j






HTI

=X

k

"(k)a†k

ak

+ const. , (2)


"(k) = 2Jp

1 2hz cos k + hz2 . (3)











hx

= 0

hz = 1

hz < 1 = (1 hz)

1/8

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx







• induces an attractive interaction between DW that for small enough can be approximated as a linear potential

Free DW

McCoy & Wu ’78



Marton Kormos




H = JLX

j=1

x

j

x

j+1

+ hzz

j

+ hxx

j

, (1)

where ↵

j






HTI

=X

k

"(k)a†k

ak

+ const. , (2)


"(k) = 2Jp

1 2hz cos k + hz2 . (3)











hx

= 0

hz = 1

hz < 1 = (1 hz)

1/8

hx

hx

V (x) = 2Jhx

|x|

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx







• induces an attractive interaction between DW that for small enough can be approximated as a linear potential

• DW do not propagate freely but get confined into mesons

Free DW Bound state = meson

McCoy & Wu ’78



Marton Kormos




H = JLX

j=1

x

j

x

j+1

+ hzz

j

+ hxx

j

, (1)

where ↵

j






HTI

=X

k

"(k)a†k

ak

+ const. , (2)


"(k) = 2Jp

1 2hz cos k + hz2 . (3)











hx

= 0

hz = 1

hz < 1 = (1 hz)

1/8

hx

hx

V (x) = 2Jhx

|x|

What happens if there are mesons in the spectrum of the post-quench Hamiltonian in the quasi-particle picture?

Back to quenches

1

t ime

space

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

ls

g

e

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

Fig. 1: Sketch...


acts as a source of quasi-particles at | 0i t = 0

Back to quenches

1

t ime

space

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

ls

g

e

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

Fig. 1: Sketch...



pairs of particles move in opposite directions with velocity vp

Back to quenches

1

t ime

space

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

ls

g

e

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

Fig. 1: Sketch...


moving away the quasi-particles feel the attractive interaction



Back to quenches

1

t ime

space

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

ls

g

e

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

Fig. 1: Sketch...


moving away the quasi-particles feel the attractive interaction

The interaction will eventually turn the particles back



Back to quenches

1

t ime

space

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

ls

g

e

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

Fig. 1: Sketch...

1-point function hx

i

Quenches from ferro to ferro

0 5 10 15 20 25 30

0.95

0.96

0.97

0.98

0.99

t

!Σ

x#

0 5 10 15 20 25 30

0.96

0.97

0.98

0.99

t

!Σ

x#

0 5 10 15 20 25 30

0.9920

0.9925

0.9930

0.9935

t

!Σ

x#

hz=0.5, hx =0, hz =0.25, hx =0.1 hz =0.5, hx =0, hz =0.25, hx =0.2 hz =0.25, hx =0, hz =0.25, hx =0.1

iTEBD vs. ED with L=8,10,12

0 0 0 0 0 0

1-point function hx

i

Quenches from ferro to ferro

0 5 10 15 20 25 30

0.95

0.96

0.97

0.98

0.99

t

!Σ

x#

0 5 10 15 20 25 30

0.96

0.97

0.98

0.99

t

!Σ

x#

0 5 10 15 20 25 30

0.9920

0.9925

0.9930

0.9935

t

!Σ

x#

hz=0.5, hx =0, hz =0.25, hx =0.1 hz =0.5, hx =0, hz =0.25, hx =0.2 hz =0.25, hx =0, hz =0.25, hx =0.1

iTEBD vs. ED with L=8,10,12

0 0 0 0 0 0

1-point function hx

i

0 1 2 3 4 5 6

0.000

0.005

0.010

0.015

Ω

!Σ#

x

"Ω#

2

0 1 2 3 4 5 6

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Ω

!Σ#

x

"Ω#

2

0 1 2 3 4 5 6

0

0.00001

0.00002

0.00003

0.00004

0.00005

Ω

!Σ#

x

"Ω#

2

Power spectrum of

m2-m1 =0.46, m1 =3.7, m2 =4.1, m3 =4.5

m2-m1 =0.68, m1 =4.0, m2 =4.7

m2-m1 =0.46, m1 =3.7, m2 =4.1, m3 =4.5

hx

i

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




Zooming in: escaping correlations

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx





2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx





0 10 20 30 40

0.00

0.02

0.04

0.06

0.08

0.10

0.12

t

!Σ

x

1Σ

x

23#

c

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




0 10 20 30

!0.00002

0

0.00002

0.00004

0.00006

t

"Σ

x

1Σ

x

23$

c


0 10 20 30 40

0.00

0.02

0.04

0.06

0.08

0.10

0.12

t

!Σ

x

1Σ

x

23#

c

Quench in the paramagnetic phase

2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx





2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




Ferro


2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




Ferro Para


2

(a)

(b)


hx

1

x

m+1

ic



hx

= 0.025, 0.05, 0.1, 0.2, 0.4.

the Hamiltonian

H = JLX

j=1

x

j

x

j+1

+ hz

z

j

+ hx

x

j

, (1)

where ↵

j



hz

and hx


ergy scale. For hx





[20, 21]:

HTFIM

=X

k

"(k)a†k

ak

+ const. , (2)


2Jp1 2h

z

cos k + hz

2.

At hz








tion = (1 hz


field hx




0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

h = 0h = 0.1h = 0.2h = 0.4

t

S

h = 2 → h = 1.75z

x

x

x

x

z

Change for small hx is perturbative.For large hx new fast excitations appear(?)No confinement.

Entanglement entropy

Ferro Para

In the Ising chain, confinement changes the light cone spreading of correlations and entanglement

Conclusions (Part II)



1

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

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a

l

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d

o

m

a

i

n

w

a

ll

s ge

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

time

space

Fig. 1: Sketch...


Questions:


1

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

ll

s ge

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

time

space

Fig. 1: Sketch...


Questions:Is it a general property of other cond-mat models featuring confinement? Presumably yes, possible to check numerically


1

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

ll

s ge

t

c

o

n

f

i

n

e

d

i

n

t

o

mes

o

n

s

time

space

Fig. 1: Sketch...


Questions:Is it a general property of other cond-mat models featuring confinement? Presumably yes, possible to check numericallyIs it true in higher dimensions? e.g. in QCD?

maybe holography can offer some hints


1

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

l

l

d

o

m

a

i

n

w

a

ll

s ge

t

c

o

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f

i

n

e

d

i

n

t

o

mes

o

n

s

time

space

Fig. 1: Sketch...

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